Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25×C4 |
| direct product | C61×#25740 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C61×#3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 81 points:
(2,3)(5,6)(7,11)(8,17)(9,15)(10,14)(12,18)(13,20)(16,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,2,4,3)(5,7,10,12)(6,11,14,18)(8,13,19,15)(9,17,20,16);(5,8,9,18,10,19,20,11)(6,12,13,17,14,7,15,16);(1,3,4,2)2623 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 18728);